The analysis of models of activity of banking structures in the conditions of perfect and imperfect competition is carried out. Production functions for financial companies are considered. Models of the bank's work as an institution of delegated monitoring are described. Models of dynamics of financial resources are analyzed, allow to describe processes of evolution of own capital of bank depending on dynamics of the involved resources and the policy of accumulation realized by it. The considered models and methods are based on the definition of a bank (a financial company) as some abstract object, which is characterized by input and output parameters, as well as the function that connects them. This approach to some extent allows adapting traditional models of research of industrial enterprises and organizations to the analysis of the activities of banking structures. Therefore, one of the main tasks is to optimize internal functioning. Under such conditions, it is especially important to consider the bank as a holistic complex dynamic system operating in an unstable transition economy and the use of economic and mathematical methods and models to study the processes taking place in the bank, assess its effectiveness, identify areas and ways to improve management banking activities. Based on the analysis of the main economic and mathematical models of behavior of financial companies in a monopolistic market, we can conclude that each model characterizes a certain aspect of financial market development by the situation. Production functions for a bank (financial firm) are built, where the problem of classification of these factors into input and output is significant. A wide class of economic and mathematical models is considered, in which the activity of financial and banking institutions is treated as financial intermediaries. The theory of delegated monitoring is generalized, which in the general case assumes that in conditions when there is an effect of the growth of income from scale, individual lenders prefer to delegate functions of control (monitoring) of the behavior of entrepreneurs in whose projects they have invested to special intermediary firms. banks. The analysis of the development of the banking sector of Ukraine showed that a further gradual slowdown in its pace is expected due to changes in conditions in global financial markets and market saturation.